Subgroups — C₂×C₆₇×#3139 #30739865

Order 17152.

2118 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3139	C₂	1	16	yes	maximal, direct product factor
4	C₂×#491×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#487×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#419×C₆₇	C₂	1	4	yes	maximal
7	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	#533×C₆₇	—	2	4	no	—
11	#533×C₆₇	C₂²	1	8	yes	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#125×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
20	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
21	C₂×#117×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#491×C₆₇	C₂²	1	32	yes	complemented
25	#487×C₆₇	C₂²	1	32	yes	complemented
26	#419×C₆₇	C₂²	1	32	yes	—
27	C₂×#103×C₆₇	C₂²	1	16	yes	—
28	C₄×M₁₆×C₆₇	C₂²	1	8	yes	—
29	C₂×#97×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
31	C₂²×Q₈×C₆₇	—	2	2	no	—
32	C₂²×Q₈×C₆₇	—	2	4	no	—
33	C₂²×Q₈×C₆₇	—	2	2	no	—
34	C₂²×Q₈×C₆₇	—	2	16	no	—
35	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
37	C₂³×C₄×C₆₇	—	2	2	no	—
38	C₂³×C₄×C₆₇	D₄	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	8	no	—
41	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₄⋊Q₈×C₆₇	—	2	16	no	—
44	C₄⋊Q₈×C₆₇	—	2	4	no	—
45	C₄⋊Q₈×C₆₇	—	2	4	no	—
46	C₄⋊Q₈×C₆₇	—	2	4	no	—
47	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
48	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
49	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
50	#125×C₆₇	—	2	4	no	—
51	#125×C₆₇	C₂³	1	8	yes	—
52	C₂²⋊Q₈×C₆₇	—	2	16	no	—
53	C₄×Q₈×C₆₇	—	2	16	no	—
54	#117×C₆₇	—	2	4	no	—
55	#117×C₆₇	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	D₄	1	2	yes	—
65	C₂×C₄²×C₆₇	D₄	1	4	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
69	#103×C₆₇	C₂³	1	64	yes	—
70	#97×C₆₇	C₂³	1	8	yes	—
71	C₄×C₈×C₆₇	C₂³	1	8	yes	—
72	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₆₇	—	2	8	no	—
74	C₂×Q₈×C₆₇	—	2	8	no	—
75	C₂×Q₈×C₆₇	—	2	4	no	—
76	C₂×Q₈×C₆₇	—	2	16	no	—
77	C₂×Q₈×C₆₇	—	2	32	no	—
78	C₂×Q₈×C₆₇	—	4	16	no	—
79	C₂×Q₈×C₆₇	—	4	8	no	—
80	C₂×Q₈×C₆₇	—	4	8	no	—
81	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₆₇	—	2	8	no	—
83	C₂²×C₄×C₆₇	—	2	8	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	4	8	no	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	#41	1	8	yes	—
93	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₆₇	—	2	16	no	—
96	C₂×C₈×C₆₇	—	2	8	no	—
97	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
98	C₄⋊C₄×C₆₇	—	2	16	no	—
99	C₄⋊C₄×C₆₇	—	2	16	no	—
100	C₄⋊C₄×C₆₇	—	2	16	no	—
101	C₄⋊C₄×C₆₇	—	2	8	no	—
102	C₄⋊C₄×C₆₇	—	2	4	no	—
103	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
104	C₂²⋊C₄×C₆₇	—	2	8	no	—
105	C₄²×C₆₇	—	2	8	no	—
106	C₄²×C₆₇	C₂×D₄	1	8	yes	—
107	C₄²×C₆₇	C₂×D₄	1	2	yes	—
108	C₄²×C₆₇	C₂×D₄	1	2	yes	—
109	C₄²×C₆₇	C₂⁴	1	2	yes	—
110	C₄²×C₆₇	C₂⁴	1	2	yes	—
111	C₂³×C₆₇	—	2	4	no	—
112	C₂³×C₆₇	#124	1	4	yes	—
113	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
114	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	16	no	—
116	Q₈×C₆₇	—	4	16	no	—
117	Q₈×C₆₇	—	4	8	no	—
118	Q₈×C₆₇	—	4	16	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	16	no	—
121	C₂×C₄×C₆₇	—	2	8	no	—
122	C₂×C₄×C₆₇	—	2	8	no	—
123	C₂×C₄×C₆₇	—	4	16	no	—
124	C₂×C₄×C₆₇	—	4	8	no	—
125	C₂×C₄×C₆₇	—	4	8	no	—
126	C₂×C₄×C₆₇	#124	1	8	yes	—
127	C₂×C₄×C₆₇	#124	1	4	yes	—
128	C₂×C₄×C₆₇	#137	1	8	yes	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
133	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
134	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
135	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
136	C₈×C₆₇	—	2	8	no	—
137	C₂²×C₆₇	—	2	8	no	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	C₂×#124	1	4	yes	—
140	C₂²×C₆₇	C₂×#124	1	2	yes	—
141	C₂²×C₆₇	C₂×#137	1	4	yes	—
142	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₆₇	—	2	4	no	—
144	C₄×C₆₇	—	4	8	no	—
145	C₄×C₆₇	—	4	8	no	—
146	C₄×C₆₇	C₂×#124	1	4	yes	—
147	C₄×C₆₇	C₂×#124	1	2	yes	—
148	C₄×C₆₇	C₂×#137	1	8	yes	—
149	C₄×C₆₇	C₂²×#41	1	2	yes	—
150	C₂×#3139	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₆₇	—	2	4	no	—
152	C₂×C₆₇	#3139	1	4	yes	direct product factor
153	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
154	C₂×C₆₇	C₂²×#137	1	2	yes	—
155	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
156	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
157	#3139	C₂×C₆₇	1	16	yes	direct product factor
158	C₂×#491	C₂×C₆₇	1	4	yes	complemented
159	C₂×#487	C₂×C₆₇	1	4	yes	complemented
160	C₂×#419	C₂×C₆₇	1	4	yes	—
161	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
162	C₆₇	C₂×#3139	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
165	#533	—	2	4	no	—
166	#533	C₂²×C₆₇	1	8	yes	—
167	C₂×C₄⋊Q₈	—	2	2	no	—
168	C₂×C₄⋊Q₈	—	2	4	no	—
169	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
171	C₂×C₄⋊Q₈	C₂²×C₆₇	1	8	yes	—
172	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
173	C₂×#125	C₂²×C₆₇	1	2	yes	—
174	C₂×C₂²⋊Q₈	—	2	2	no	—
175	C₂×C₄×Q₈	—	2	2	no	—
176	C₂×#117	C₂²×C₆₇	1	2	yes	—
177	C₂²×C₄⋊C₄	—	2	2	no	—
178	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
179	#491	C₂²×C₆₇	1	32	yes	complemented
180	#487	C₂²×C₆₇	1	32	yes	complemented
181	#419	C₂²×C₆₇	1	32	yes	—
182	C₂×#103	C₂²×C₆₇	1	16	yes	—
183	C₄×M₁₆	C₂²×C₆₇	1	8	yes	—
184	C₂×#97	C₂²×C₆₇	1	2	yes	—
185	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
186	C₂²×Q₈	—	2	2	no	—
187	C₂²×Q₈	—	2	2	no	—
188	C₂²×Q₈	—	2	4	no	—
189	C₂²×Q₈	—	2	16	no	—
190	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
191	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	D₄×C₆₇	1	2	yes	—
194	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
195	C₂×M₁₆	—	2	8	no	—
196	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
197	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
198	C₄⋊Q₈	—	2	16	no	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
203	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
204	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
205	#125	—	2	4	no	—
206	#125	C₂³×C₆₇	1	8	yes	—
207	C₂²⋊Q₈	—	2	16	no	—
208	C₄×Q₈	—	2	16	no	—
209	#117	—	2	4	no	—
210	#117	C₂³×C₆₇	1	8	yes	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	16	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	D₄×C₆₇	1	2	yes	—
220	C₂×C₄²	D₄×C₆₇	1	4	yes	—
221	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
224	#103	C₂³×C₆₇	1	64	yes	—
225	#97	C₂³×C₆₇	1	8	yes	—
226	C₄×C₈	C₂³×C₆₇	1	8	yes	—
227	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	16	no	—
232	C₂×Q₈	—	2	32	no	—
233	C₂×Q₈	—	4	16	no	—
234	C₂×Q₈	—	4	8	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
237	C₂²×C₄	—	2	8	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	8	no	—
242	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
244	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
247	C₂²×C₄	#41×C₆₇	1	8	yes	—
248	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
250	M₁₆	—	2	16	no	—
251	C₂×C₈	—	2	8	no	—
252	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	16	no	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₄²	—	2	8	no	—
261	C₄²	C₂×D₄×C₆₇	1	8	yes	—
262	C₄²	C₂×D₄×C₆₇	1	2	yes	—
263	C₄²	C₂×D₄×C₆₇	1	2	yes	—
264	C₄²	C₂⁴×C₆₇	1	2	yes	—
265	C₄²	C₂⁴×C₆₇	1	2	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	#124×C₆₇	1	4	yes	—
268	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
269	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
270	Q₈	—	2	16	no	—
271	Q₈	—	4	16	no	—
272	Q₈	—	4	8	no	—
273	Q₈	—	4	16	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	16	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	4	16	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	#124×C₆₇	1	8	yes	—
282	C₂×C₄	#124×C₆₇	1	4	yes	—
283	C₂×C₄	#137×C₆₇	1	8	yes	—
284	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
285	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
286	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
288	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
289	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
290	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
291	C₈	—	2	8	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	8	no	—
294	C₂²	C₂×#124×C₆₇	1	4	yes	—
295	C₂²	C₂×#124×C₆₇	1	2	yes	—
296	C₂²	C₂×#137×C₆₇	1	4	yes	—
297	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	4	8	no	—
300	C₄	—	4	8	no	—
301	C₄	C₂×#124×C₆₇	1	4	yes	—
302	C₄	C₂×#124×C₆₇	1	2	yes	—
303	C₄	C₂×#137×C₆₇	1	8	yes	—
304	C₄	C₂²×#41×C₆₇	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₆₇×#3139	1	4	yes	direct product factor
307	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
308	C₂	C₂²×#137×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₆₇×#3139	C₂	direct (N × H)
4. C₂×#491×C₆₇	C₂	semidirect
5. C₂×#487×C₆₇	C₂	semidirect
24. #491×C₆₇	C₂²	semidirect
25. #487×C₆₇	C₂²	semidirect
150. C₂×#3139	C₆₇	direct (N × H)
152. C₂×C₆₇	#3139	direct (N × H)
157. #3139	C₂×C₆₇	direct (N × H)
158. C₂×#491	C₂×C₆₇	semidirect
159. C₂×#487	C₂×C₆₇	semidirect
162. C₆₇	C₂×#3139	direct (N × H)
179. #491	C₂²×C₆₇	semidirect
180. #487	C₂²×C₆₇	semidirect
306. C₂	C₆₇×#3139	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#533×C₆₇	2
C₂²×C₄⋊Q₈×C₆₇	2
C₆₇×#3139	2
C₂×#491×C₆₇	2
C₂×#487×C₆₇	2
C₂×#419×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#3139	67

Subgroups — C2×C67×#3139 #30739865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3139 #30739865